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:toc:
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[[chp:opalcycl]]
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:stem: latexmath
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:sectnums:
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[[opal---madx-conversion-guide]]
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_OPAL_ - MADX Conversion Guide
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------------------------------
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We note with latexmath:[\alpha],latexmath:[\beta] and
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latexmath:[\gamma] the Twiss parameters.
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[latexmath]
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++++
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\begin{aligned}
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\sigma_{beam} &=& \begin{pmatrix}\sigma_{x} & \sigma_{x p_{x}}\\\sigma_{x p_{x}} & \sigma_{ p_{x}}\end{pmatrix}
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= \begin{pmatrix} \sigma_{x} & \delta\cdot\sqrt{\sigma_{x}\sigma_{ p_{x}}}\\\delta\cdot\sqrt{\sigma_{x}\sigma_{ p_{x}}} & \sigma_{ p_{x}}\end{pmatrix}
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= \begin{pmatrix} <x^{2}> & <x p_{x}>\\<x p_{x}> & < p_{x}^{2}> \end{pmatrix} \\
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&=& \begin{pmatrix}
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\frac{1}{N}\sum_{i=1}^{N}x_{i}^{2} & \frac{1}{N}\sum_{i=1}^{N}x_{i} p_{x_{i}}\\
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\frac{1}{N}\sum_{i=1}^{N}x_{i} p_{x_{i}} & \frac{1}{N}\sum_{i=1}^{N} p_{x_{i}}^{2}
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\end{pmatrix}
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= \epsilon\cdot\begin{pmatrix} \beta & -\alpha\\ -\alpha & \gamma\end{pmatrix}
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\end{aligned}
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++++
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[latexmath]
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++++
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\begin{aligned}
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\bar{p}_{x} & = & \sqrt{\frac{1}{N}\sum_{i=1}^{N} p_{x_{i}}^{2}} & = & \sqrt{\sigma_{ p_{x}}} & & \bar{x} & = & \sqrt{\frac{1}{N}\sum_{i=1}^{N}x_{i}^{2}} \\
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\bar{p}_{y} & = & \sqrt{\frac{1}{N}\sum_{i=1}^{N}p_{y_{i}}^{2}} & = & \sqrt{\sigma_{p_{y}}} & & \bar{y} & = & \sqrt{\frac{1}{N}\sum_{i=1}^{N}y_{i}^{2}} \\
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\\
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\\
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\gamma & = & \frac{E_{kin}+m_{p}}{m_{p}} & & & & \beta & = & \sqrt{1-\frac{1}{\gamma^{2}}} = \frac{v}{c} \\
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(\beta\gamma) & = & \frac{E_{kin}+m_{p}}{m_{p}}\cdot\sqrt{1-\frac{1}{\gamma^{2}}}& = & \frac{\beta}{\sqrt{1-\beta^{2}}} & & \text{B}\rho & = & \frac{\left(\beta\gamma\right)\cdot m_{p}\cdot 10^{9}}{c}~\left[\text{T m}\right] \\
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m_{p} & = & 0.939277 [GeV] & & & & c & = & 299792458 [m/s]
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\end{aligned}
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++++
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[cols="<,^,>,<,^,<,^,>",options="header",]
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|=======================================================================
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|Quantity |`MADX` | | |Conversion | |_OPAL-Output_ |
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|Momenta |latexmath:[\bar{p}_{x}] |[rad]
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|latexmath:[\bar{p}_{x}][latexmath:[\beta\gamma]] |=
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|latexmath:[\left(\bar{p}_{x}\left[\text{rad}\right]\right)\cdot\left(\beta\gamma\right)]
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|latexmath:[\bar{p}_{x}] |[latexmath:[\beta\gamma]]
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|Correlation of latexmath:[\bar{x}],latexmath:[\bar{p}_{x}]
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|latexmath:[\delta] |[1] |latexmath:[\delta] |=
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|latexmath:[\left(\sigma_{x p_{x}}\left[\text{m }\text{rad}\right]\right)/\left(\left(\bar{p}_{x}\left[\text{rad}\right]\right)\cdot\left(
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\bar{x}\left[\text{m}\right]\right)\right)] |latexmath:[\delta] |[1]
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| | | | |=
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|latexmath:[\left(\sigma_{x p_{x}}\left[\text{m }\text{rad}\right]\right)/\sqrt{\left(\sigma_{x}\left[\text{m}^{2}\right]\right)\cdot\left(\sigma_{ p_{x}}\left[\text{rad}^{2}\right]\right)}]
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|Emittance |latexmath:[\epsilon_{x}] |[m rad]
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|latexmath:[\epsilon_{x}][m latexmath:[\beta\gamma]] |=
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|latexmath:[\sqrt{\left( \bar{p}_{x}\left[ \beta\gamma \right] \right) ^{2} \cdot \left(\bar{x}\left[\text{m}\right]\right)^{2} - \left(\delta \cdot \left(\bar{x}\left[\text{m}\right]\right) \cdot \left(\bar{p}_{x}\left[\beta\gamma\right]\right)\right)^{2}} ]
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|latexmath:[\epsilon_{x}] |[m latexmath:[\beta\gamma]]
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| | | | |=
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|latexmath:[\sqrt{\left( \sigma_{{p}_{x}}\left[ \left(\beta\gamma\right)^{2} \right] \right)\cdot
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\left(\sigma_{x}\left[\text{m}^{2}\right]\right) - \left(\delta \cdot
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\sqrt{\left(\sigma_{x}\left[\text{m}^{2}\right]\right)\cdot\left(\sigma_{p_{x}}
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\left[\left(\beta\gamma\right)^{2}\right]\right)}\right)^{2}} ] | |
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|latexmath:[\sqrt{\left( \sigma_{{p}_{x}}\left[ \left(\beta\gamma\right)^{2} \right] \right)\cdot
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\left(\sigma_{x}\left[\text{m}^{2}\right]\right) - \left(\sigma_{x p_{x}}\left[\text{m} ~\beta\gamma\right]\right)^{2}} ]
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|Twiss Parameter latexmath:[\alpha] |latexmath:[\alpha] |[1]
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|latexmath:[\alpha\left[1\right]] |=
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|latexmath:[-\delta\cdot\left(\bar{x}\left[\text{m}\right]\right)\cdot\left(\bar{p}_{x}\left[\beta\gamma\right]\right)/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]\right)]
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|latexmath:[\alpha_{T}] |[1]
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| | | | |=
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|latexmath:[-\delta\cdot\sqrt{\left(\sigma_{x}\left[\text{m}^{2}\right]\right)\cdot\left(\sigma_{ p_{x}}\left[\left(\beta\gamma\right)^{2}\right]\right)}/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]\right)]
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|Twiss Parameter latexmath:[\beta_{T}] |latexmath:[\beta_{T}]
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|[m/rad] |latexmath:[\beta_{T}\left[\text{m}/\beta\gamma\right]] |=
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|latexmath:[\left(\bar{x}\left[\text{m}\right]\right)^{2}/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]
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\right)] |latexmath:[\beta_{T}] |[m/latexmath:[\beta\gamma]]
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|latexmath:[\left(\sigma_{x}\left[\text{m}^{2}\right]\right)/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]
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\right)] | |
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|Twiss Parameter latexmath:[\gamma_{T}] |latexmath:[\gamma_{T}]
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|[rad/m] |latexmath:[\gamma_{T}\left[\beta\gamma/\text{m}\right]] |=
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|latexmath:[\left(\bar{p}_{x}\left[\beta\gamma\right]\right)^{2}/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]\right)]
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|latexmath:[\gamma_{T}] |[latexmath:[\beta\gamma]/m]
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| | | | |=
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|latexmath:[\left(\sigma_{p_{x}}\left[\left(\beta\gamma\right)^{2}\right]\right)/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]\right)]
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|Focusing strength |latexmath:[k_{1}]
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|latexmath:[\left[\text{m}^{-2}\right]]
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|latexmath:[k_{1}\left[\text{T}/\text{m}\right]] |=
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|latexmath:[\left(k_{1}\left[\text{m}^{-2}\right]\right)\cdot\left(\text{B}\rho\left[\text{T m}\right]\right)]
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|latexmath:[k_{1}] |latexmath:[\left[\text{T}/\text{m}\right]]
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|Quantity |`MADX` | | |Conversion | |_OPAL-Input_ |
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|Element Position |`at :=` |latexmath:[\left[\text{m}\right]]
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|`ELEMEDGE` |= |(Center of the element) - (Length of the element)/2
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|`ELEMEDGE =` |latexmath:[\left[\text{m}\right]]
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| |Center of the element | | | | |Begin of the element |
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|Quantity |_OPAL-Output_ | | |Conversion | |_OPAL-Input_ |
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|Momenta |latexmath:[\bar{p}_{x}]
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|latexmath:[\left[\beta\gamma\right]]
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|latexmath:[p_{x}\left[\text{eV}\right]] |=
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|latexmath:[m_{p}\cdot10^{9}\cdot\left(\sqrt{\left(\bar{p}_{x}\left[\beta\gamma\right]\right)^{2} +1}-1\right)]
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|latexmath:[\bar{p}_{x}] |[eV]
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|======================================================================= |